Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models

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January 2010, 13(1): 79-100. doi: 10.3934/dcdsb.2010.13.79

Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models

Marco Di Francesco ^1, and Donatella Donatelli ^2,

1.	Sezione di Matematica per l’Ingegneria, Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Facoltà di Ingegneria, Piazzale E. Pontieri, I-67040 - Monteluco di Roio - L’Aquila, Italy
2.	Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell'Aquila, Via Vetoio, 67010 Coppito (AQ)

Received January 2008 Revised December 2008 Published October 2009

Abstract
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In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller-Segel type systems. The approximating systems are either hyperbolic-parabolic or hyperbolic-elliptic. They all feature a nonlinear pressure term arising from a volume filling effect which takes into account the fact that cells do not interpenetrate. The main convergence result relies on energy methods and compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two-dimensional case with periodical boundary conditions.

Keywords: Keller Segel model, chemotaxis, diffusive relaxation, Singular convergence, volume filling effect., nonlinear diffusion, nonlinear hyperbolic systems.

Mathematics Subject Classification: Primary: 35L99, 35B25, 92C17; Secondary: 35Q80, 76N1.

Citation: Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79

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